A symmetry is a change which returns the system to an state indistinguishable for its starting point. For example, the rotations of n120º or reflections, flips, across each altitude of an equilateral triangle. For each transformation in the table, the starting point is the original triangle (123).

original(123)

rotated 120º cw(312)

rotated 120º ccw(231)

reflected across 1(132)

reflected across 2(321)

reflected across 3(213)

Notice that they all retain the shape of an equilateral triangle.

Or perhaps we could example the rotations of n90º or reflections, flips, across the perpendicular bisectors of the top/bottom and sides or reflections across the diagonal bisectors of the angles of a square.

original(1234)

rotate cw 90º(4123)

rotate cw 180º(3412)

rotated cw 270º(2341)

reflection acrossvertical bisector(2143)

reflection across horizontal bisector(4321)

reflection acrossleft-right diagonal(1432)

reflection across opposite diagonal(3214)

Notice that all these transformations still retain the shape of a square.

Without going through the constructions, the reader should see that the transmutations resulting from rotating a circle through any angle, or any fraction of an angle, about an axis passing through its center will result in another circle indistinguishable from the original.

Our first two examples involving equilateral triangles and squares are called discrete symmetries; while, this last example is called a continuous symmetry.

Noether's Theorems

Amalie Noether (1882-1935) proved one of the greatest relationships between mathematics and physics - For every continuous symmetry in nature there is a universal conservation law of physics. And, for every conservation law in nature there must exist a continuous symmetry.

Continuous symmetry under time translation is equivalent to the law of conservation of energy.

Continuous symmetry under space translation is equivalent to the law of conservation of linear momentum.

Continuous symmetry under rotation is equivalent to the law of conservation of angular momentum.

Most students are familiar with "shape symmetries" from their study of geometry. This is not the same as the mathematical symmetry Noether is discussing. Although, shape symmetries are important in physics and often allow students the opportunity to gain valuable information regarding the solution to a problem, we are now going to examine Noether's concept of symmetry..

Symmetries in mathematics are studied in group theory, usually offered in courses called modern or abstract algebra. If objects form a group they must obey the following properties:

The group <G, *> is closed under an operation, *. This means that the result of the operation * on two members of the group produces another member of the group.

There is an identity element, e, in <G, *> such that a * e = e * a = a

Every member of the group, <G, *>, has a unique inverse; that is, a * a-1 = a-1 * a = e.

If a group also obeys the commutative property, a * b = b * a for all elements in the group then it is called an abelian group.

Using the equilateral triangle transformations in the previous section, we can show that they form a group by using a Cayley table or magic square. In order to fill out the table, we need to give titles to each of our transformations.

identity, e (123)

rotating 120º cw (312)

rotating 120º ccw (231)

reflection across blue (23)

reflection across green (13)

reflection across red (12)

o

e

(312)

(231)

(23)

(13)

(12)

e

e

(312)

(231)

(23)

(13)

(12)

(312)

(312)

(231)

e

(12)

(23)

(13)

(231)

(231)

e

(312)

(13)

(12)

(23)

(23)

(23)

(23)

(12)

e

(312)

(231)

(13)

(13)

(12)

(23)

(231)

e

(312)

(12)

(12)

(23)

(13)

(312)

(231)

e

Notice that this group is closed under the binary operation, o. That is, every combination, a o b, results in an element of the original set. Notice that there is an identity element, (123) or e. As seen in the table, every element has a unique inverse; that is, the identity, e, appears only once is each row and column. This group can be shown to be associative by examining (a o b) o c = a o (b o c) for any three elements in the set. This group is not commutative; that is, the order of the operation (row o column) does not always give the same result as (column o row).

Many group symmetries have special names that all mathematicians and physicists recognize, such as, SO(N) [rotations of a circle SO(2) or a sphere SO(3)], SU(N), Sp(N), and the exceptional groups G2, F4, E6, E7, and E8. The above group of transformations on an equilateral triangle is called S3. These are discussions that are well beyond the scope of introductory physics. But it is noteworthy to mention that the following internal symmetries are often referred to when studying the basis for the standard model of particle theory, U(1) x SU(2) x SU(3).

What are some examples of groups in mathematics?

When studying number theory, there are special groups that most students will recognize. One group is the integers under addition, <Z, +>. Remember that the integers can be listed as the set containing the positive and negative whole numbers with the addition of zero, or {… -3, -2, -1 0 1, 2, 3,….}. It can be easily verified that this is a group under addition by noticing the presence of the identity element zero, the fact that the set is closed since any two integers when added together give another integer, the fact that addition is associative, and that every positive integer has a unique additive inverse (its negative integer) which when added together equal zero. This group is also commutative.

But would the set of only positive integers, Z+, be a group under the binary operation of addition? Or the set of only negative integers, Z-, be a group under the binary operation of addition? The answer is no for either collection, or set, since each respectively has no identity element or additive inverses.

Note that the set of integers in not a group under the binary operation of multiplication since there are no multiplicative inverses (1/p is not an integer). Despite the fact that the integers are closed under multiplication, multiplication obeys the associative property, and the integers have an identity element, 1. The integers are also not a group under division since division is not associative: (2/5)/10 = 0.4 / 10 = 0.04 while 2/(5/10) = 2 / 0.5 = 4.

Notice that the set of rational numbers, Q (any number that can be expressed as a quotient of two integers as long as the denominator is not equal to zero), is a group under any of the binary operations of multiplication, addition, and subtraction. But why not under division? The answer is, once again, that division is not associative.

In mathematics, a field is a commutative group with respect to two operations. For example, the rational numbers, Q, are a field under addition and multiplication since they are closed under both operations, have identity elements and inverses under both operations, associative under both operations, obey the distributive property of multiplication over addition, a(b + c) = ab + ac. The real numbers (rationals plus irrationals) and complex numbers (a + bi) also form fields under addition and multiplication.

Fields in Physics

In physics, a field can be thought of as a property of space. For example, Newton's gravitational field is a region surrounding a body with mass in which another body having mass will experience a force of attraction. This field follows an inverse square function and is thought to have non-zero values everywhere. Faraday later described an electric field as a region surrounding a charged body (a point charge) in which another charged object will experience a force of attraction or repulsion. This field also follows an inverse square function and is thought to have non-zero values everywhere.

These fields assign a value to each position in space with respect to the location of the center mass/charge. With respect to gravitation, the value is called the gravitational field strength (g = GM/r2) and with respect to electric fields (E = kQ/r2). Since these quantities have magnitude and a direction, these fields are called vector fields. These fields have space symmetry since they can be translated or rotated in space and still have the same relative values.

Maxwell's electromagnetic field is actually two vector fields (electric and magnetic). This field has a "physical reality" because they carry energy (electromagnetic radiation). Similar to how the electric field between parallel plates of a capacitor has an energy density, or energy per unit volume, of u = ½(eoE2). Similarly, the magnetic field within a solenoid has a energy density of u = ½(B2/µo). One of the most extraordinary results of Maxwell's work was uncovering the fact that the speed of light, c, is equal to

Potential, or voltage, is an example of a scalar field with respect to a particular location in space. To find the potential at a point in space, all you have to do is sum up the contribution of potential due to each nearby charge (VP = kq1/r1P + kq2/r2P + ….). This result is true regardless of the exact position of the charges in space, just as long as their relative positions to each other remain the same. So it is symmetric under translations and rotations of space.

Now let's elaborate a little further on vector fields.

Since the gravitational and electric forces experienced by masses or charges placed in either of these fields are themselves vectors, we will now define a vector space consisting of arrows in a plane(s). This vector space is a set of vectors, V, over a scalar field F of real numbers together with two binary operations: vector addition and scalar multiplication. Our vector space must satisfy the following axioms:

vector addition must be closed - that is, adding two vectors A and B must result in a third vector that is also in the plane

there must be an identity element for vector addition - that is, a vector of length zero, 0 = <0i, 0j>

A note about notation. Sometimes it is more convenient to discuss a vector, not as a diagram, but as an ordered pair. On a two-dimensional plane, like those diagrammed above, the magnitude of the vector can be written in terms of the unit vectors i = <1,0> and j = <0,1>. So a vector that is created by moving 3 units in the x-directions (horizontal) and then 4 units in the y-directions (vertical) could be expressed as u = <3i, 4j>. this vector would have a magnitude of 5 and a direction of 37º. Writing the vector as u = (5, 37º) is called polar notation, while writing it as <3i, 4j> is called component notation.

there must be an additive inverse for each vector - that is, A + (-A) = 0

The 0 vector can be written as 0 = <0i, 0j>. -A is merely the vector having the same magnitude as A but oriented in a direction that is revered by 180º.

scalar multiplication must be compatible with field multiplication - that is, x(yA) = (xy)A

3(< 4i, 4j >) = 12< i, j >

< 12i, 12j > = 12< i, j >

there must be an identity element for scalar multiplication - that is, 1A = A where 1 is the multiplicative identity of the field

Notice that all of these axioms just boil down to the fact that a vector space is a mathematical field and once that has been stated, none of the above proofs would be necessary. That is, the mathematical field properties are the properties of the physical vector space. Consequently we now know that these properties can be automatically applied to any vector space whether it be vectors representing displacement, velocity, acceleration, forces, electric fields, magnetic field strength, gravitational field strength, etc. They are universal and can be used as a basis of working with vectors in any order spacetime.

Summary

The benefit of mapping a mathematical structure to a physical process allows theorists the ability to test and predict. For example, in beta decay, physicists in the early 1900's originally feared that momentum and energy might not be conserved. They thought that the conservation laws might have reached their limit of applicability. But that would mean that there were limitations on continuous symmetries under time and space translations. That concept was even more frightening. When Pauli suggested a new particle, the neutrino, that would have zero charge and a small amount of mass, then beta decay would then be composed of a neutron decaying into a proton, electron, and a neutrino. The neutrino would carry away the missing momentum and energy not present in the observed proton and electron. Today physicists know that there are actually six different types of neutrinos: electron neutrinos, muon neutrinos, tau neutrinos, and their associated antiparticles. Pauli's rigid belief in symmetry and its associated conservation processes introduced a whole new class of particles.

Mathematics is the language of physics.

Leon Lederman and Chris Hill, in their book Symmetry and the Beautiful Universe, state in pages 65-67 that

"Nature has deep underpinnings, nature speaks through the language of mathematics. A theoretical mathematician attempts to create a roadmap of all possible logical systems that could consistently exist, whether they ultimately have anything to do with nature's design or not. … While a theoretical physicist, often borrows from mathematics (or, if there is nothing to borrow, they invent new mathematics) in order to create a mathematical roadmap of things in the real world, in nature. …. Nature gives math its inspiration … but math does not have to do experimentation in the physical world … a logical, consistent, abstract formulation is sufficient. Noether has given us the connection between these two disciples in her theorems."

Richard Feynman in his 1964 Messenger Series of lectures at Cornell, which were later printed in his book The Character of Physical Law, designated two cultures:

"those who know and understand mathematics well enough to see the beauty of nature and those who have not. ... It is too bad that it has to be mathematics, and that mathematics is hard for some people.... Physicists cannot make a conversion to any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. ... In the same way all the intellectual arguments in the world will not convey an understanding of nature to those of 'the other culture.' Philosophers may try to teach you by telling you qualitatively about nature, [physicists are] trying to describe her." (page 58)